Separations in Proof Complexity and TFNP

• DOI: 10.1145/3663758
• Publication Date: 2024-05-09T08:33:50Z
• Authors: Mika Göös; Alexandros Hollender; Siddhartha Jain; Gilbert Maystre; William Pires; Robert Robere; Ran Tao
• Keywords: FOS: Computer and information sciences; Computer Science - Computational Complexity; 0102 computer and information sciences; 0101 mathematics; Computational Complexity (cs.CC); 01 natural sciences

It is well-known that Resolution proofs can be efficiently simulated by Sherali–Adams (SA) proofs. We show, however, that any such simulation needs to exploit huge coefficients: Resolution cannot be efficiently simulated by SA when the coefficients are written in unary. We also show that Reversible Resolution (a variant of MaxSAT Resolution) cannot be efficiently simulated by Nullstellensatz (NS).

These results have consequences for total NP search problems. First, we characterise the classes PPADS, PPAD, SOPL by unary-SA, unary-NS, and Reversible Resolution, respectively. Second, we show that, relative to an oracle,

\({\text{ PLS}} \not\subseteq {\text{ PPP}}\)

,

\({\text{ SOPL}} \not\subseteq {\text{ PPA}}\)

, and

\({\text{ EOPL}} \not\subseteq {\text{ UEOPL}}\)

. In particular, together with prior work, this gives a complete picture of the black-box relationships between all classical TFNP classes introduced in the 1990s.